Flory-Schulz fraction distribution

Figure 10.3. Schulz-Flory mole-fraction distribution (left) and corresponding molecular-weight distribution (right) at different degrees of fractional conversion of functional groups (adapted from Flory [27]).

Free-radical polymerization with chain breaking exclusively by disproportionation or chain transfer yielding unreactive radicals produces a Schulz-Flory mole-fraction distribution. 

The normal distribution function, also referred to as the Flory-Schulz distribution, relates the fraction of an x-mer (a polymer molecule consisting of x repeat units) in the entire assembly of molecules to its formation probability. It can be defined either as a number distribution function or as a weight distribution function. The number of moles of an x-mer (Nx) is given by the normal number distribution as follows ... 

A mole-fraction distribution that is a declining geometrical progression is called a Schulz-Flory distribution or most probable distribution and is quite common [29,30]. As later examples will show, it can arise from other mechanisms as well and can therefore not be taken as evidence for step growth. 

Flory Schulz MWD, The weight fraction of polymer at each degree of polymerization, x, at extent of reaction, p, described by the Flory-Scnulz distribution 13, 14) is given by... 

Figure 1. Wesslau and Flory-Schulz differential weight-fraction MWDs on a logarithmic scale, where W is the weight fraction and M is the molecular weight. Both distributions are for M = 10,000 gjmol and M /M = 2.0.

The molar mass distributiMi for two polymers. The first polymer (continuous trace) follows the Flory-Schulz MMD with Mn = 40000 Mw is the double, i.e., Mw = 80000). The second polymer (dotted trace) follows the Schulz-Zimm MMD with Mn = 40000 and Mw = 120000. The molar mass distribution is diplayed as the molar fraction (a) and weight fraction (b) vs. mass. 

The aforementioned expression is the geometric distribution or the Flory-Schulz distribution. The results can be illustrated by plotting the mole fraction of chain length for different values of conversion, p. 

When determining the product selectivities, all compounds of equal carbon numbers (paraffines, olefins, isomers, and oxygen compounds) were summarized to one product fraction. The chain growth probability was determined by the Anderson-Schulz-Flory (ASF) distribution ... 

The description of the product distribution for an FT reaction can be simplified and described by the use of a single parameter (a value) determined from the Anderson-Schulz-Flory (ASF) plots. The a value (also called the chain growth probability factor) is then used to describe the total product spectrum in terms of carbon number weight fractions during the FT synthesis. In the case... 

The carbon number distribution of Fischer-Tropsch products on both cobalt and iron catalysts can be clearly represented by superposition of two Anderson-Schulz-Flory (ASF) distributions characterized by two chain growth probabilities and the mass or molar fraction of products assigned to one of these distributions.7 10 In particular, this bimodal-type distribution is pronounced for iron catalysts promoted with alkali (e.g., K2C03). Comparing product distributions obtained on alkali-promoted and -unpromoted iron catalysts has shown that the distribution characterized by the lower growth probability a, is not affected by the promoter, while the growth probability a2 and the mass fraction f2 are considerably increased by addition of alkali.9 This is... 

Zirconia-modified silica impregnated with Co2(CO)s and activated under H2 at 300 °C renders a catalyst more active and selective to diesel fraction, in the CO hydrogenation reaction, than that conventionally prepared from a nitrate salt solution. The selectivity patterns followed a Schulz-Flory distribution and catalysts prepared from carbonyl precursor exhibited low water-gas shift activity [146]. 

The experimentally obtained Anderson-Schulz-Flory (ASF) distribution (solid line) follows the theoretical values closely and was an early indication that the reaction to form the hydrocarbons was a type of polymerization, and indeed of Ci species. An interesting feature of the ASF plot is that it is not quite smooth but has a kink at A = 2 which comes below the curve (see Figure 15). The reason why substantially less ethane and ethylene than expected is formed has been widely debated it can occur if fewer free C2 species are produced or if the C2 fraction preferentially undergoes further reaction. The former explanation seems to be the more accepted one, in other words the rate at which surface-attached C2 undergoes further polymerization is faster than the rate of liberation of the free C2 hydrocarbons from the surface. 

In a recent study, R. Pettit et at. examined the validity of tire Fischer-Tropsch carbide mechanism, the Anderson-Emmett hydroxy carbene mechanism and the Pichlcr-Schulz mediaiiism [174. In a first experiment, the Schulz Flory distribution obtained by CO/H conversion over a cobalt catalyst in the absence and in the presence of CH N] was studied. It was found that addition of CHjN resulted in a signillcant increase of the propagation rate which is in favour of the assumption of methylene as a building block, as predicted by the carbide mechanism. Furthermore, the reaction was carried out using labeled CO (90% CO and 10% CO), H2. and CHjNj in variable ratios. The number of atoms in the propenc fraction was calculated according to the three... 

Given the Schulz-Flory distribution, the mole and weight fractions of polymer with j monomer units (based on polymer) as a function of that number j are... 

In all the above ethylene oligomerizations, Schulz-Flory distributions of oUgomers are observed which can be quantified by the a value, which represents the probability of chain propagation [117, 118]. The magnitude of a can be experimentally determined by the ratio of two oligomer fractions Eq. (5.1). The a value is usually preferred to be in the range of about 0.6 to about 0.8 to make a-olefins of the most commercial interest... 

Overall hydrocarbon distributions on the four carbon supports are shown in Figure 11. The hydrocarbon mole fractions comply with the Anderson-Schulz-Flory (ASF) distribution. Note that hydrocarbons of carbon number up to 34 are detected for all carbon supports. However, the chain-growth probability (a) for catalyst with wood-AC support is 0.65 (at least for carbon numbers of up to 20), smaller than that of the catalysts using the other three AC supports ( 0.71). 

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